There have been concerns that Wednesday's announcement by Ehud Olmert, the Israeli prime minister, that he would not run in his Kadima party's leadership elections in September following corruption scandals could hamper progress in the talks.
However Saeb Erekat, the Palestinian negotiator who was also in Washington for the talks, said that all sides would continue to push for an agreement.
"We decided today [Wednesday] that we are going to continue pursuing to reach an agreement before the end of the year," Erekat said after the meetings.
Sean McCormack, the US state department spokesman, also said on Wednesday following Olmert's announcement that all parties were "going to keep moving forward" and that the administration would work with all "responsible" Israeli leaders.
"The fact of the matter is, if you wait for the perfect moment to try to help bring the two parties together in a final agreement, you're going to be waiting forever," he said.
The Israeli prime minister suggested earlier this week that there was little chance of reaching the end of the year deadline as it would be impossible to agree on the status of Jerusalem - parts of which Palestinians claim as the capital of any future Palestinian state.
He had said efforts to reach agreement over borders and refugees could be bridged.
Qurei had said on Tuesday after separate talks with Rice at the US state department that there would be "no agreement" without Jerusalem.
Both sides committed to forging a comprehensive deal by the end of 2008 during talks in Annapolis, Maryland, in November last year.
Rice is also expected to travel to the Middle East soon in a bid to speed up efforts on the talks.
Reports say the US secretary of state is keen to get both sides to agree on a document of understanding on key issues ahead of the UN General Assembly session in September.
McCormack would not confirm the reports, AFP said.